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受欢迎的三角函数 >

3tan^2(x)+5tan(x)+2=0

解答

3 tan^{2} (x) + 5 tan (x) + 2 = 0

解答

x = - 0.58800 \dots + πn, x = \frac{3 π}{4} + πn

+1

度数

x = - 33.69006 \dots^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

求解步骤

3 tan^{2} (x) + 5 tan (x) + 2 = 0

用替代法求解

3 tan^{2} (x) + 5 tan (x) + 2 = 0

令

: tan (x) = u

3 u^{2} + 5 u + 2 = 0

3 u^{2} + 5 u + 2 = 0 : u = - \frac{2}{3}, u = - 1

3 u^{2} + 5 u + 2 = 0

使用求根公式求解

3 u^{2} + 5 u + 2 = 0

二次方程求根公式:

若

a = 3, b = 5, c = 2

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 \cdot 2}{2 \cdot 3}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 \cdot 2}{2 \cdot 3}

5^{2} - 4 \cdot 3 \cdot 2 = 1

5^{2} - 4 \cdot 3 \cdot 2

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 5^{2} - 24

5^{2} = 25

= 25 - 24

数字相减:

25 - 24 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 5 \pm 1}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- 5 + 1}{2 \cdot 3}, u_{2} = \frac{- 5 - 1}{2 \cdot 3}

u = \frac{- 5 + 1}{2 \cdot 3} : - \frac{2}{3}

\frac{- 5 + 1}{2 \cdot 3}

数字相加/相减:

- 5 + 1 = - 4

= \frac{- 4}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 4}{6}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{6}

约分:

2

= - \frac{2}{3}

u = \frac{- 5 - 1}{2 \cdot 3} : - 1

\frac{- 5 - 1}{2 \cdot 3}

数字相减:

- 5 - 1 = - 6

= \frac{- 6}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 6}{6}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{6}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = - \frac{2}{3}, u = - 1

u = tan (x)

代回

tan (x) = - \frac{2}{3}, tan (x) = - 1

tan (x) = - \frac{2}{3}, tan (x) = - 1

tan (x) = - \frac{2}{3} : x = arctan (- \frac{2}{3}) + πn

tan (x) = - \frac{2}{3}

使用反三角函数性质

tan (x) = - \frac{2}{3}

tan (x) = - \frac{2}{3}

的通解

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{2}{3}) + πn

x = arctan (- \frac{2}{3}) + πn

tan (x) = - 1 : x = \frac{3 π}{4} + πn

tan (x) = - 1

tan (x) = - 1

的通解

tan (x)

周期表（周期为

πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

合并所有解

x = arctan (- \frac{2}{3}) + πn, x = \frac{3 π}{4} + πn

以小数形式表示解

x = - 0.58800 \dots + πn, x = \frac{3 π}{4} + πn

作图

流行的例子

sin(2x+pi/3)=-(sqrt(3))/2

sin (2 x + \frac{π}{3}) = - \frac{3}{2}

tan(2θ)+2cos(θ)=0,0<= θ<= 2pi

tan (2 θ) + 2 cos (θ) = 0, 0 \leq θ \leq 2 π

cos(2x)+10sin^2(x)=5

cos (2 x) + 10 sin^{2} (x) = 5

2 = 3 + 2 sin (θ)

-2sin(θ)+sin(2θ)=0

- 2 sin (θ) + sin (2 θ) = 0